percentage base problem solving

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How to Solve Percent Problems? (+FREE Worksheet!)

Learn how to calculate and solve percent problems using the percent formula.

Related Topics

• How to Find Percent of Increase and Decrease
• How to Find Discount, Tax, and Tip
• How to Do Percentage Calculations
• How to Solve Simple Interest Problems

Step by step guide to solve percent problems

• In each percent problem, we are looking for the base, or part or the percent.
• Use the following equations to find each missing section. Base \(= \color{black}{Part} \ ÷ \ \color{blue}{Percent}\) \(\color{ black }{Part} = \color{blue}{Percent} \ ×\) Base \(\color{blue}{Percent} = \color{ black }{Part} \ ÷\) Base

Percent Problems – Example 1:

\(2.5\) is what percent of \(20\)?

In this problem, we are looking for the percent. Use the following equation: \(\color{blue}{Percent} = \color{ black }{Part} \ ÷\) Base \(→\) Percent \(=2.5 \ ÷ \ 20=0.125=12.5\%\)

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Percent problems – example 2:.

\(40\) is \(10\%\) of what number?

Use the following formula: Base \(= \color{ black }{Part} \ ÷ \ \color{blue}{Percent}\) \(→\) Base \(=40 \ ÷ \ 0.10=400\) \(40\) is \(10\%\) of \(400\).

Percent Problems – Example 3:

\(1.2\) is what percent of \(24\)?

In this problem, we are looking for the percent. Use the following equation: \(\color{blue}{Percent} = \color{ black }{Part} \ ÷\) Base \(→\) Percent \(=1.2÷24=0.05=5\%\)

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Percent problems – example 4:.

\(20\) is \(5\%\) of what number?

Use the following formula: Base \(= \color{black}{Part} \ ÷ \ \color{blue}{Percent}\) \(→\) Base \(=20÷0.05=400\) \( 20\) is \(5\%\) of \(400\).

Exercises for Calculating Percent Problems

Solve each problem..

• \(51\) is \(340\%\) of what?
• \(93\%\) of what number is \(97\)?
• \(27\%\) of \(142\) is what number?
• What percent of \(125\) is \(29.3\)?
• \(60\) is what percent of \(126\)?
• \(67\) is \(67\%\) of what?

Download Percent Problems Worksheet

• \(\color{blue}{15}\)
• \(\color{blue}{104.3}\)
• \(\color{blue}{38.34}\)
• \(\color{blue}{23.44\%}\)
• \(\color{blue}{47.6\%}\)
• \(\color{blue}{100}\)

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Finding the Base Number in a Percent Problem Worksheet

Related Topics & Worksheets: Reverse Percentage Percentage Worksheet

Objective: I can find the base number in a percent problem.

Example: 8 is 32% of what number?

Answer: 25  

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Solving problems with percentages

• Price difference I
• Price difference II
• How many students?

To solve problems with percent we use the percent proportion shown in "Proportions and percent".

$$\frac{a}{b}=\frac{x}{100}$$

$$\frac{a}{{\color{red} {b}}}\cdot {\color{red} {b}}=\frac{x}{100}\cdot b$$

$$a=\frac{x}{100}\cdot b$$

x/100 is called the rate.

$$a=r\cdot b\Rightarrow Percent=Rate\cdot Base$$

Where the base is the original value and the percentage is the new value.

47% of the students in a class of 34 students has glasses or contacts. How many students in the class have either glasses or contacts?

$$a=r\cdot b$$

$$47\%=0.47a$$

$$=0.47\cdot 34$$

$$a=15.98\approx 16$$

16 of the students wear either glasses or contacts.

We often get reports about how much something has increased or decreased as a percent of change. The percent of change tells us how much something has changed in comparison to the original number. There are two different methods that we can use to find the percent of change.

The Mathplanet school has increased its student body from 150 students to 240 from last year. How big is the increase in percent?

We begin by subtracting the smaller number (the old value) from the greater number (the new value) to find the amount of change.

$$240-150=90$$

Then we find out how many percent this change corresponds to when compared to the original number of students

$$90=r\cdot 150$$

$$\frac{90}{150}=r$$

$$0.6=r= 60\%$$

We begin by finding the ratio between the old value (the original value) and the new value

$$percent\:of\:change=\frac{new\:value}{old\:value}=\frac{240}{150}=1.6$$

As you might remember 100% = 1. Since we have a percent of change that is bigger than 1 we know that we have an increase. To find out how big of an increase we've got we subtract 1 from 1.6.

$$1.6-1=0.6$$

$$0.6=60\%$$

As you can see both methods gave us the same answer which is that the student body has increased by 60%

Video lessons

A skirt cost $35 regulary in a shop. At a sale the price of the skirtreduces with 30%. How much will the skirt cost after the discount?

Solve "54 is 25% of what number?"

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Percent , Rate , Base

Understanding percent , rate , and base is essential in various mathematical and real-life contexts. In this study guide, we will cover the basics of percent , rate , and base , and provide examples to help you grasp these concepts.

Percent means "per hundred" and is denoted by the symbol "%". It is used to express a number as a fraction of 100. For example, 25% is equivalent to the fraction 25/100 or the decimal 0.25.

A rate is a special ratio in which the two terms are in different units . For example, miles per hour (mph) is a rate . It compares the distance traveled to the time taken. Rates are often expressed using the word "per" or the symbol "/", such as 60 miles per hour or 60 mph.

The base is the original value in a percent problem. It is the whole or the original amount before a percentage is calculated. For example, if you're calculating 20% of 80, then 80 is the base .

Key Formulas

The following formulas are essential when dealing with percent , rate , and base :

Percent = (Part / Whole) * 100

Rate = (Part / Base )

Base = (Part / Rate )

Let's work through a few examples to illustrate these concepts:

Example 1: Calculating Percent

If you scored 35 out of 50 on a test, what is your score as a percentage?

Percent = (35 / 50) * 100 = 70%

Example 2: Calculating Rate

If a car travels 300 miles in 5 hours , what is its speed in miles per hour ?

Rate = 300 miles / 5 hours = 60 mph

Example 3: Finding the Base

If 15 is 20% of a number, what is the original number?

Base = 15 / 0.20 = 75

When studying percent , rate , and base , it's helpful to practice converting between fractions , decimals , and percentages . Additionally, working through real-life problems involving discounts, taxes, and tips can improve your understanding of these concepts.

Remember to use the key formulas and units to guide your problem-solving process. Understanding the relationship between percent , rate , and base will also make it easier to solve problems in various scenarios.

By mastering percent , rate , and base , you'll develop a valuable skill set for handling a wide range of mathematical and practical situations.

Good luck with your studies!

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• Percent word problem: recycling cans

Percent word problems

• Rates and percentages FAQ
• Your answer should be
• an integer, like 6 ‍  
• a simplified proper fraction, like 3 / 5 ‍  
• a simplified improper fraction, like 7 / 4 ‍  
• a mixed number, like 1   3 / 4 ‍  
• an exact decimal, like 0.75 ‍  
• a multiple of pi, like 12   pi ‍   or 2 / 3   pi ‍  

Percent Maths Problems

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Definition:

The percent, base, and rate are connected with one another in terms of computation. To find the percentage , multiply the base by the rate. Remember that the rate must be changed from a percent to a decimal before multiplying can be done. Rate times base equals percentage.

PERCENTAGE (P=BxR) –  The result obtained when a number is multiplied by a percent.

BASE (B=P/R) –  The whole in a problem. The amount you are taking a percent of.

RATE (R=P/B) –  The ratio of amount to the base. It is written as a percent.

Applying Percentage, Base, and Rate Worksheets

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Solving Percent Problems

Percent is a great mathematical tool to express quantities and is used extensively in different things – from interest rates, discounts, and taxes to surveys, censuses, etc.

This article is your guide to percent and solving percent problems frequently appearing in major national examinations.

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Table of Contents

What does percent mean.

The word “percent” originated from the Latin phrase per centum, meaning “by hundred.” When we say “percent,” we refer to “parts per 100”. This means that a percent is a fraction with 100 as the denominator. The symbol % is used to indicate a percent.

For example, 3% means three parts per 100 or 3⁄100; 45% means 45⁄100; and 92% means 92⁄100.

Illustrating Percent

Suppose a vendor has 100 biscuits. If 10% of those biscuits are ube-flavored, 10⁄100 or 10 out of 100 biscuits are ube-flavored.

On the other hand, suppose there are 100 students in a school auditorium. If 42% of those students are honor students, 42⁄100 students, or 42 out of 100 students, are honor students.

Expressing Percent as Fraction and Decimal

Since percent means a fraction with 100 as the denominator, we can express a percent as a fraction or a decimal number .

Drop the percent sign and put 100 as the denominator to transform a percent into a fraction. For instance, 25% is simply 25⁄100.

Note that when 25⁄100 is reduced to its lowest terms, you will obtain ¼. This means that 25% is also equivalent to ¼. 

Furthermore, note that when you transform ¼ into its decimal form using the steps we have discussed in the previous reviewer , you will obtain 0.25. Hence, 25% is also equal to 0.25. 

There is an easier way to transform percent into decimals . Drop the percent sign and move the decimal point two places to the left of the given number.

For example, 54% is equivalent to 0.54

Example: Transform 3% to decimal form.

Suppose that your mom prepared ten pieces of your favorite cookies. You are excited to taste those cookies, but you realize that your brother ate 20% of the cookies that your mom prepared. What exactly is the number of cookies eaten by your brother?

To determine the answer to your question above, you must determine 20% of 10. This case involves the application of percentages.

The percentage is the result when you multiply a number by a percent. Returning to your problem about the number of cookies your brother ate, 20% of 10 can be determined if you multiply ten by 20%. The result after you multiply the numbers is called the percentage.

How To Find the Percentage

Follow these steps if you want to find the percentage:

Step 1: Convert the given percent (the one with the % sign) into decimals .

Again, to convert percent into its decimal form, we drop the percent sign and then move the decimal point two places to the left. Thus, 20% = 0.20

Step 2 : Multiply the decimal you have obtained from Step 1 to the given number. The result is the percentage.

To multiply 0.20 by 10, we ignored the decimal point for a while and multiplied the given decimals like whole numbers. We have obtained 0200. Since 0.20 has two decimal places while 10 has none, the final answer should have two decimal places. We count two digits from the right of 0200 and put the decimal point there. Hence, the answer is 02.00, which is equivalent to 2.

Hence, 20% of 10 is 2. This means that out of 10 cookies your mother prepared, 2 of those were eaten by your brother.

Let us have another example.

Example: What is 50% of 120?

Step 1 : Convert the given percent (the one with the % sign) into decimals.

We drop the % sign of 50% and move the decimal point two places to the left.

Thus, 50% = 0.50

Step 2: Multiply the decimal you have obtained from Step 1 to the given number. The result is the percentage.

To multiply 0.50 by 120, we ignored the decimal point for a while and multiplied the given decimals like whole numbers. Through this process, we have obtained 06000. Since 0.50 has two decimal places while 120 has none, the final answer should have two decimal places. We count two digits from the right of 06000 and put the decimal point there. Hence, the answer is 060.00, which is equivalent to 60.

Hence, 50% of 120 is 60.

Simple Tricks in Computing Percentages

We always want to make our computations in mathematics faster and more accurate. For this reason, I will share two tricks you can use when computing percentages.

Trick #1: You can compute some percentages using only mental computation.

If you want to determine the 25%, 50%, 75%, or 100% of a number, you can do so without the help of pen and paper.

• 25% is equivalent to 25⁄100 or ¼. Hence, to find the 25% of a number, divide the given number by 4. Example: 25% of 40 is just 40 ÷ 4 = 10.
• 50% is equivalent to 50⁄100 or ½. Thus, to find the 50% of a number, divide the given number by 2. This means 50% of a number is just half the given number. Example: 50% of 40 is just 40 ÷ 2 = 20.
• 75% is equivalent to 75⁄100 or ¾. Thus, to find the 75% of a number, multiply the given number by three and then divide the result by 4. Example: 75% of 40 is just 40 x 3 = 120 ÷ 4 = 30.
• 100% is equivalent to 100⁄100 or 1. Thus, 100% of a number is the number itself . Example: 100% of 40 is just 40 itself.

Trick #2: X% of a number Y is equal to Y% of number X

This trick means we can transfer the % sign to the other number, and the result will be the same.

Example : What is 40% of 25?

Using trick #2, we can transfer the % sign from 40% to 25. Thus, we have 25%. This means 40% of 25 is the same as 25% of 40.

Thus, applying our first trick on finding the 25% of a number, 40 ÷ 4 = 10; hence, 40% of 25 is 10.

Example : What is 92% of 50? 

92% of 50 is the same as 50% of 92. Hence, we can just divide 92 by 2 to obtain the answer, 92 ÷ 2 = 46

Therefore, 92% of 50 is 46.

Base and Rate

The base is the amount you are taking a percent of. Meanwhile, the rate is the percent you are calculating.

For example, if there are 50 students in a classroom and 20% of those students are honor students, it follows that ten students are honor students. 50 is the base since it is the amount we take a percent of. Meanwhile, 20% is the rate since we calculate the percentage. Lastly, 10 is the percentage.

The product of the base and the rate is the percentage .

Percentage = Base × Rate

Example: Determine the percentage, base, and rate if 20% of 90 is 18.

Since 90 x 20% = 90 x 0.20 = 18, 90 is the base, 20% is the rate, and 18 is the percentage.

Calculating Percentage, Base, and Rate

Formula to find the percentage.

The formula to find the percentage, as we have stated, is: 

We can manipulate the mathematical equation above to obtain the formulas for computing the base and the rate:

Formula to Find the Base

Base = Percentage ÷ Rate

Formula to Find the Rate

Rate = Percentage ÷ Base

Example 1: If 10% of a number is 90, what is the number?

We can interpret this question as 10% of ______ = 90. Since “of” is a signal word for multiplication, it also implies 10% x ______ = 90

This means that 10% is the rate while 90 is the percentage. The unknown number is the base. Thus, we need to compute the base.

Using the formula to find the base:

  Base = Percentage ÷ Rate

Base  = 90 ÷ 10%

Convert the given percent into decimal:

Base  = 90 ÷ 0.10

Now that you have already transformed the rate into decimal form, you may divide 90 by 0.10 to obtain the answer.

To perform division with decimal numbers , we need to transform the divisor (0.10) into a whole number by moving two decimal places to the right. Thus, the new divisor is 10. We also move two decimal places for the dividend (90). Thus, the new dividend is 9000.

We now perform long division with our new dividend and divisor:

To find the base, we compute 90 ÷ 0.10 = 900

Hence, the base is 900.

Example 2:  What percent of 720 is 90?

We can translate the question above in this form: _____% of 720 is 90 or _____% x 720 = 90. Therefore, 720 is the base, while 90 is the percentage. The missing number is the rate.

We will now use the formula for finding the rate.

Again, based on the given problem, the percentage is 90 while the base is 720

          Rate = 90 ÷ 720

Notice that the dividend (the first number) is smaller than the divisor (the second number). In this case, you may apply the same steps in transforming fractions into decimal form because  90 ÷ 720 is a proper fraction (i.e., 90⁄720).

Let us divide 90 by 720 using the steps in transforming fractions into decimal form .

We add some zeros and decimal points to proceed with the division process.

We can now divide 900 by 720.

Note that every time the remainder becomes smaller than the divisor, we add zeros to 900 and the remainder to continue the division process.

The quotient we obtained is 0.125. Thus, 0.125 is our rate.

However, the rate must always be expressed with a percent sign. To do this, we multiply 0.125 by 100 or move two decimal places to the right of it and put a percent sign. Thus, 0.125 is equal to 12.5%.

Therefore, the rate is 12.5%

The Percentage, Base, and Rate Triangle

What if you forgot the formula to determine the percentage, base, or rate in a particular problem? Don’t worry because there is a fun way to derive these formulas. 

Shown below is the Percentage, Base, and Rate Triangle . It is a triangle divided into three portions where P (for percentage) is written on the upper portion, and B (for base) and R (for rate) are written on the lower portions. There are also division signs in the triangle’s outer left and outer right parts and a multiplication sign below it.

How To Use the Percentage, Base, and Rate Triangle

Suppose you are looking for the base. You have to cover the B in the triangle and look at the remaining letters and the operation between them. Notice that if you cover B, the remaining letters are P and R, with a division sign between them. This means that to find the base, you must divide P by R.

Next topic:  Ratio and Proportion

Previous topic : Fundamental Operations on Fractions and Decimals

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Jewel Kyle Fabula

Jewel Kyle Fabula is a Bachelor of Science in Economics student at the University of the Philippines Diliman. His passion for learning mathematics developed as he competed in some mathematics competitions during his Junior High School years. He loves cats, playing video games, and listening to music.

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18.4: Solving Percentage Problems

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Let's solve more percentage problems.

Exercise \(\PageIndex{1}\): Number Talk: Multiplication with Decimals

Find the products mentally.

\(6\cdot (0.8)\cdot 2\)

\((4.5)\cdot (0.6)\cdot 4\)

Exercise \(\PageIndex{2}\): Coupons

Han and Clare go shopping, and they each have a coupon. Answer each question and show your reasoning.

• Han buys an item with a normal price of $15, and uses a 10% off coupon. How much does he save by using the coupon?

• Clare buys an item with a normal price of $24, but saves $6 by using a coupon. For what percentage off is this coupon?

Are you ready for more?

Clare paid full price for an item. Han bought the same item for 80% of the full price. Clare said, “I can’t believe I paid 125% of what you paid, Han!” Is what she said true? Explain.

Exercise \(\PageIndex{3}\): Info Gap: Music Devices

Your teacher will give you either a problem card or a data card . Do not show or read your card to your partner.

If your teacher gives you the problem card :

• Silently read your card and think about what information you need to be able to answer the question.
• Ask your partner for the specific information that you need.
• Explain how you are using the information to solve the problem. Continue to ask questions until you have enough information to solve the problem.
• Share the problem card and solve the problem independently.
• Read the data card and discuss your reasoning.

If your teacher gives you the data card :

• Silently read your card.
• Ask your partner “What specific information do you need?” and wait for them to ask for information. If your partner asks for information that is not on the card, do not do the calculations for them. Tell them you don’t have that information.
• Before sharing the information, ask “ Why do you need that information? ” Listen to your partner’s reasoning and ask clarifying questions.
• Read the problem card and solve the problem independently.
• Share the data card and discuss your reasoning.

A pot can hold 36 liters of water. What percentage of the pot is filled when it contains 9 liters of water?

Here are two different ways to solve this problem:

• Using a double number line:

We can divide the distance between 0 and 36 into four equal intervals, so 9 is \(\frac{1}{4}\) of 36, or 9 is 25% of 36.

• Using a table:

Glossary Entries

Definition: Percent

The word percent means “for each 100.” The symbol for percent is %.

For example, a quarter is worth 25 cents, and a dollar is worth 100 cents. We can say that a quarter is worth 25% of a dollar.

Definition: Percentage

A percentage is a rate per 100.

For example, a fish tank can hold 36 liters. Right now there is 27 liters of water in the tank. The percentage of the tank that is full is 75%.

Exercise \(\PageIndex{4}\)

For each problem, explain or show your reasoning.

• 160 is what percentage of 40?
• 40 is 160% of what number?
• What number is 40% of 160?

Exercise \(\PageIndex{5}\)

A store is having a 20%-off sale on all merchandise. If Mai buys one item and saves $13, what was the original price of her purchase? Explain or show your reasoning.

Exercise \(\PageIndex{6}\)

The original price of a scarf was $16. During a store-closing sale, a shopper saved $12 on the scarf. What percentage discount did she receive? Explain or show your reasoning.

Exercise \(\PageIndex{7}\)

Select all the expressions whose value is larger than 100.

• 120% of 100
• 500% of 400
• 1% of 1,000

Exercise \(\PageIndex{8}\)

An ant travels at a constant rate of 30 cm every 2 minutes.

• At what pace does the ant travel per centimeter?
• At what speed does the ant travel per minute?

(From Unit 3.3.4)

Exercise \(\PageIndex{9}\)

Is \(3\frac{1}{2}\) cups more or less than 1 liter? Explain or show your reasoning. (Note: 1 cup \(\approx\) 236.6 milliliters)

(From Unit 3.2.3)

Exercise \(\PageIndex{10}\)

Name a unit of measurement that is about the same size as each object.

• The distance of a doorknob from the floor is about 1 _____________.
• The thickness of a fingernail is about 1 _____________.
• The volume of a drop of honey is about 1 _____________.
• The weight or mass of a pineapple is about 1 _____________.
• The thickness of a picture book is about 1 _____________.
• The weight or mass of a buffalo is about 1 _____________.
• The volume of a flower vase is about 1 _____________.
• The weight or mass of 20 staples is about 1 _____________.
• The volume of a melon is about 1 _____________.
• The length of a piece of printer paper is about 1 _____________.

(From Unit 3.2.1)

Grade 6 Mathematics Module: Finding the Percentage, Base and Rate in a Given Problem

This Self-Learning Module (SLM) is prepared so that you, our dear learners, can continue your studies and learn while at home. Activities, questions, directions, exercises, and discussions are carefully stated for you to understand each lesson.

Each SLM is composed of different parts. Each part shall guide you step-by-step as you discover and understand the lesson prepared for you.

Pre-tests are provided to measure your prior knowledge on lessons in each SLM. This will tell you if you need to proceed on completing this module or if you need to ask your facilitator or your teacher’s assistance for better understanding of the lesson. At the end of each module, you need to answer the post-test to self-check your learning. Answer keys are provided for each activity and test. We trust that you will be honest in using these.

Please use this module with care. Do not put unnecessary marks on any part of this SLM. Use a separate sheet of paper in answering the exercises and tests. And read the instructions carefully before performing each task.

If you have any questions in using this SLM or any difficulty in answering the tasks in this module, do not hesitate to consult your teacher or facilitator.

This module was designed and written with you in mind. It is here to help you master the lessons on Finding the Percentage, Base and Rate or Percent in given problems. The scope of this module permits it to be used in many different learning situations. The language used recognizes your diverse vocabulary level. The lessons are arranged to follow the standard sequence of the course. But the order in which you read them can be changed to correspond with the textbook you are now using.

The module is divided into three lessons, namely:

• Lesson 1 – Finding the Percentage in a Given Problem
• Lesson 2 – Finding the Rate in a Given Problem
• Lesson 3 – Finding the Base in a Given Problem

After going through this module, you are expected to:

1. identify the percentage, rate and base in a given problem;

2. find the base, percentage or rate or percent in a given problem; and

3. solve routine and non-routine problems involving the percentage, rate and base using appropriate strategies and tools.

Grade 6 Mathematics Quarter 2 Self-Learning Module: Finding the Percentage, Base and Rate in a Given Problem

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Determining Percentage, Base, & Rate

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This serves as additional worksheet to develop skill in identifying and solving for the percentage, base or rate in a given situation

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Watch CBS News

Teens come up with trigonometry proof for Pythagorean Theorem, a problem that stumped math world for centuries

By Bill Whitaker

May 5, 2024 / 7:00 PM EDT / CBS News

As the school year ends, many students will be only too happy to see math classes in their rearview mirrors. It may seem to some of us non-mathematicians that geometry and trigonometry were created by the Greeks as a form of torture, so imagine our amazement when we heard two high school seniors had proved a mathematical puzzle that was thought to be impossible for 2,000 years. 

We met Calcea Johnson and Ne'Kiya Jackson at their all-girls Catholic high school in New Orleans. We expected to find two mathematical prodigies.

Instead, we found at St. Mary's Academy , all students are told their possibilities are boundless.

Come Mardi Gras season, New Orleans is alive with colorful parades, replete with floats, and beads, and high school marching bands.

In a city where uniqueness is celebrated, St. Mary's stands out – with young African American women playing trombones and tubas, twirling batons and dancing - doing it all, which defines St. Mary's, students told us.

Junior Christina Blazio says the school instills in them they have the ability to accomplish anything. 

Christina Blazio: That is kinda a standard here. So we aim very high - like, our aim is excellence for all students. 

The private Catholic elementary and high school sits behind the Sisters of the Holy Family Convent in New Orleans East. The academy was started by an African American nun for young Black women just after the Civil War. The church still supports the school with the help of alumni.

In December 2022, seniors Ne'Kiya Jackson and Calcea Johnson were working on a school-wide math contest that came with a cash prize.

Ne'Kiya Jackson: I was motivated because there was a monetary incentive.

Calcea Johnson: 'Cause I was like, "$500 is a lot of money. So I-- I would like to at least try."

Both were staring down the thorny bonus question.

Bill Whitaker: So tell me, what was this bonus question?

Calcea Johnson: It was to create a new proof of the Pythagorean Theorem. And it kind of gave you a few guidelines on how would you start a proof.

The seniors were familiar with the Pythagorean Theorem, a fundamental principle of geometry. You may remember it from high school: a² + b² = c². In plain English, when you know the length of two sides of a right triangle, you can figure out the length of the third.

Both had studied geometry and some trigonometry, and both told us math was not easy. What no one told  them  was there had been more than 300 documented proofs of the Pythagorean Theorem using algebra and geometry, but for 2,000 years a proof using trigonometry was thought to be impossible, … and that was the bonus question facing them.

Bill Whitaker: When you looked at the question did you think, "Boy, this is hard"?

Ne'Kiya Jackson: Yeah. 

Bill Whitaker: What motivated you to say, "Well, I'm going to try this"?

Calcea Johnson: I think I was like, "I started something. I need to finish it." 

Bill Whitaker: So you just kept on going.

Calcea Johnson: Yeah.

For two months that winter, they spent almost all their free time working on the proof.

CeCe Johnson: She was like, "Mom, this is a little bit too much."

CeCe and Cal Johnson are Calcea's parents.

CeCe Johnson:   So then I started looking at what she really was doing. And it was pages and pages and pages of, like, over 20 or 30 pages for this one problem.

Cal Johnson: Yeah, the garbage can was full of papers, which she would, you know, work out the problems and-- if that didn't work she would ball it up, throw it in the trash. 

Bill Whitaker: Did you look at the problem? 

Neliska Jackson is Ne'Kiya's mother.

Neliska Jackson: Personally I did not. 'Cause most of the time I don't understand what she's doing (laughter).

Michelle Blouin Williams: What if we did this, what if I write this? Does this help? ax² plus ….

Their math teacher, Michelle Blouin Williams, initiated the math contest.

Bill Whitaker: And did you think anyone would solve it?

Michelle Blouin Williams: Well, I wasn't necessarily looking for a solve. So, no, I didn't—

Bill Whitaker: What were you looking for?

Michelle Blouin Williams: I was just looking for some ingenuity, you know—

Calcea and Ne'Kiya delivered on that! They tried to explain their groundbreaking work to 60 Minutes. Calcea's proof is appropriately titled the Waffle Cone.

Calcea Johnson: So to start the proof, we start with just a regular right triangle where the angle in the corner is 90°. And the two angles are alpha and beta.

Bill Whitaker: Uh-huh

Calcea Johnson: So then what we do next is we draw a second congruent, which means they're equal in size. But then we start creating similar but smaller right triangles going in a pattern like this. And then it continues for infinity. And eventually it creates this larger waffle cone shape.

Calcea Johnson: Am I going a little too—

Bill Whitaker: You've been beyond me since the beginning. (laughter) 

Bill Whitaker: So how did you figure out the proof?

Ne'Kiya Jackson: Okay. So you have a right triangle, 90° angle, alpha and beta.

Bill Whitaker: Then what did you do?

Ne'Kiya Jackson: Okay, I have a right triangle inside of the circle. And I have a perpendicular bisector at OP to divide the triangle to make that small right triangle. And that's basically what I used for the proof. That's the proof.

Bill Whitaker: That's what I call amazing.

Ne'Kiya Jackson: Well, thank you.

There had been one other documented proof of the theorem using trigonometry by mathematician Jason Zimba in 2009 – one in 2,000 years. Now it seems Ne'Kiya and Calcea have joined perhaps the most exclusive club in mathematics. 

Bill Whitaker: So you both independently came up with proof that only used trigonometry.

Ne'Kiya Jackson: Yes.

Bill Whitaker: So are you math geniuses?

Calcea Johnson: I think that's a stretch. 

Bill Whitaker: If not genius, you're really smart at math.

Ne'Kiya Jackson: Not at all. (laugh) 

To document Calcea and Ne'Kiya's work, math teachers at St. Mary's submitted their proofs to an American Mathematical Society conference in Atlanta in March 2023.

Ne'Kiya Jackson: Well, our teacher approached us and was like, "Hey, you might be able to actually present this," I was like, "Are you joking?" But she wasn't. So we went. I got up there. We presented and it went well, and it blew up.

Bill Whitaker: It blew up.

Calcea Johnson: Yeah. 

Ne'Kiya Jackson: It blew up.

Bill Whitaker: Yeah. What was the blowup like?

Calcea Johnson: Insane, unexpected, crazy, honestly.

It took millenia to prove, but just a minute for word of their accomplishment to go around the world. They got a write-up in South Korea and a shout-out from former first lady Michelle Obama, a commendation from the governor and keys to the city of New Orleans. 

Bill Whitaker: Why do you think so many people found what you did to be so impressive?

Ne'Kiya Jackson: Probably because we're African American, one. And we're also women. So I think-- oh, and our age. Of course our ages probably played a big part.

Bill Whitaker: So you think people were surprised that young African American women, could do such a thing?

Calcea Johnson: Yeah, definitely.

Ne'Kiya Jackson: I'd like to actually be celebrated for what it is. Like, it's a great mathematical achievement.

Achievement, that's a word you hear often around St. Mary's academy. Calcea and Ne'Kiya follow a long line of barrier-breaking graduates. 

The late queen of Creole cooking, Leah Chase , was an alum. so was the first African-American female New Orleans police chief, Michelle Woodfork …

And judge for the Fifth Circuit Court of Appeals, Dana Douglas. Math teacher Michelle Blouin Williams told us Calcea and Ne'Kiya are typical St. Mary's students.  

Bill Whitaker: They're not unicorns.

Michelle Blouin Williams: Oh, no no. If they are unicorns, then every single lady that has matriculated through this school is a beautiful, Black unicorn.

Pamela Rogers: You're good?

Pamela Rogers, St. Mary's president and interim principal, told us the students hear that message from the moment they walk in the door.

Pamela Rogers: We believe all students can succeed, all students can learn. It does not matter the environment that you live in. 

Bill Whitaker: So when word went out that two of your students had solved this almost impossible math problem, were they universally applauded?

Pamela Rogers: In this community, they were greatly applauded. Across the country, there were many naysayers.

Bill Whitaker: What were they saying?

Pamela Rogers: They were saying, "Oh, they could not have done it. African Americans don't have the brains to do it." Of course, we sheltered our girls from that. But we absolutely did not expect it to come in the volume that it came.  

Bill Whitaker: And after such a wonderful achievement.

Pamela Rogers: People-- have a vision of who can be successful. And-- to some people, it is not always an African American female. And to us, it's always an African American female.

Gloria Ladson-Billings: What we know is when teachers lay out some expectations that say, "You can do this," kids will work as hard as they can to do it.

Gloria Ladson-Billings, professor emeritus at the University of Wisconsin, has studied how best to teach African American students. She told us an encouraging teacher can change a life.

Bill Whitaker: And what's the difference, say, between having a teacher like that and a whole school dedicated to the excellence of these students?

Gloria Ladson-Billings: So a whole school is almost like being in Heaven. 

Bill Whitaker: What do you mean by that?

Gloria Ladson-Billings: Many of our young people have their ceilings lowered, that somewhere around fourth or fifth grade, their thoughts are, "I'm not going to be anything special." What I think is probably happening at St. Mary's is young women come in as, perhaps, ninth graders and are told, "Here's what we expect to happen. And here's how we're going to help you get there."

At St. Mary's, half the students get scholarships, subsidized by fundraising to defray the $8,000 a year tuition. Here, there's no test to get in, but expectations are high and rules are strict: no cellphones, modest skirts, hair must be its natural color.

Students Rayah Siddiq, Summer Forde, Carissa Washington, Tatum Williams and Christina Blazio told us they appreciate the rules and rigor.

Rayah Siddiq: Especially the standards that they set for us. They're very high. And I don't think that's ever going to change.

Bill Whitaker: So is there a heart, a philosophy, an essence to St. Mary's?

Summer Forde: The sisterhood—

Carissa Washington: Sisterhood.

Tatum Williams: Sisterhood.

Bill Whitaker: The sisterhood?

Voices: Yes.

Bill Whitaker: And you don't mean the nuns. You mean-- (laughter)

Christina Blazio: I mean, yeah. The community—

Bill Whitaker: So when you're here, there's just no question that you're going to go on to college.

Rayah Siddiq: College is all they talk about. (laughter) 

Pamela Rogers: … and Arizona State University (Cheering)

Principal Rogers announces to her 615 students the colleges where every senior has been accepted.

Bill Whitaker: So for 17 years, you've had a 100% graduation rate—

Pamela Rogers: Yes.

Bill Whitaker: --and a 100% college acceptance rate?

Pamela Rogers: That's correct.

Last year when Ne'Kiya and Calcea graduated, all their classmates went to college and got scholarships. Ne'Kiya got a full ride to the pharmacy school at Xavier University in New Orleans. Calcea, the class valedictorian, is studying environmental engineering at Louisiana State University.

Bill Whitaker: So wait a minute. Neither one of you is going to pursue a career in math?

Both: No. (laugh)

Calcea Johnson: I may take up a minor in math. But I don't want that to be my job job.

Ne'Kiya Jackson: Yeah. People might expect too much out of me if (laugh) I become a mathematician. (laugh)

But math is not completely in their rear-view mirrors. This spring they submitted their high school proofs for final peer review and publication … and are still working on further proofs of the Pythagorean Theorem. Since their first two …

Calcea Johnson: We found five. And then we found a general format that could potentially produce at least five additional proofs.

Bill Whitaker: And you're not math geniuses?

Bill Whitaker: I'm not buying it. (laughs)

Produced by Sara Kuzmarov. Associate producer, Mariah B. Campbell. Edited by Daniel J. Glucksman.

Bill Whitaker is an award-winning journalist and 60 Minutes correspondent who has covered major news stories, domestically and across the globe, for more than four decades with CBS News.

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Two-Stage Attention Model to Solve Large-Scale Traveling Salesman Problems

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• Feng Wang 12 &
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The Traveling Salesman Problem (TSP) widely exists in real-world scenarios. Various methods, such as exact methods, heuristic methods, and deep learning-based methods, can solve TSPs efficiently. However, as the size of the problems increases, these methods become increasingly time-consuming due to the high complexity of large-scale TSPs. This paper proposes a two-stage attention model (TSAM) that incorporates the divide-and-conquer strategy and attention model to solve large-scale TSPs efficiently. Experimental results demonstrate that TSAM can rapidly produce promising solutions for TSP instances ranging from 500 to 10,000 nodes.

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Acknowledgment

This work is supported by the National Nature Science Foundation of China [Grant Nos. 62173258, 61773296].

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School of Computer Science, Wuhan University, Wuhan, China

Qi He & Feng Wang

Hongyi Honor College, Wuhan University, Wuhan, China

Jingge Song

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Correspondence to Feng Wang .

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Zhejiang University, Hangzhou, China

Zheng-Guang Wu

Guangdong University of Technology, Guangzhou, China

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He, Q., Wang, F., Song, J. (2024). Two-Stage Attention Model to Solve Large-Scale Traveling Salesman Problems. In: Luo, B., Cheng, L., Wu, ZG., Li, H., Li, C. (eds) Neural Information Processing. ICONIP 2023. Lecture Notes in Computer Science, vol 14448. Springer, Singapore. https://doi.org/10.1007/978-981-99-8082-6_10

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Published : 15 November 2023

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